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Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices

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  • Ningning Xia

    (Shanghai University of Finance and Economics)

  • Zhidong Bai

    (Northeast Normal University)

Abstract

In this paper, we adopt the eigenvector empirical spectral distribution (VESD) to investigate the limiting behavior of eigenvectors of a large dimensional Wigner matrix $$\mathbf {W}_n.$$ W n . In particular, we derive the optimal bound for the rate of convergence of the expected VESD of $$\mathbf{W}_n$$ W n to the semicircle law, which is of order $$O(n^{-1/2})$$ O ( n - 1 / 2 ) under the assumption of having finite 10th moment. We further show that the convergence rates in probability and almost surely of the VESD are $$O(n^{-1/4})$$ O ( n - 1 / 4 ) and $$O(n^{-1/6}),$$ O ( n - 1 / 6 ) , respectively, under finite eighth moment condition. Numerical studies demonstrate that the convergence rate does not depend on the choice of unit vector involved in the VESD function, and the best possible bound for the rate of convergence of the VESD is of order $$O(n^{-1/2}).$$ O ( n - 1 / 2 ) .

Suggested Citation

  • Ningning Xia & Zhidong Bai, 2019. "Convergence rate of eigenvector empirical spectral distribution of large Wigner matrices," Statistical Papers, Springer, vol. 60(3), pages 983-1015, June.
    • Handle: RePEc:spr:stpapr:v:60:y:2019:i:3:d:10.1007_s00362-016-0860-x
    DOI: 10.1007/s00362-016-0860-x
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    References listed on IDEAS

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    Cited by:

    1. Li, Yuling & Zhou, Huanchao & Hu, Jiang, 2023. "The eigenvector LSD of information plus noise matrices and its application to linear regression model," Statistics & Probability Letters, Elsevier, vol. 197(C).

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